In this talk we consider the ring of equivariant cohomology of moment-angle complex. We discuss how to compute it and talk about properties of this ring, in particular, we take a look at conditions under which the equivariant cohomology ring is a free module over equivariant cohomology ring of a point.

Torus actions on symplectic toric manifolds and contact toric manifolds share certain similar properties, so-called “local standards”. This allows us to recover the original manifold from the orbit space together with torus action data. We discuss certain wide class of manifolds which includes both of above manifolds and study their equivariant classification.

- 소스 관리 툴의 대세, 해외 IT 시장에서는 Git이 기본, 모르는 자는 취업조차 어렵다. - 프로그램 코드와 연구를 하나의 툴로 관리하자 - 나만의 Git 저장소를 만드는 법 - 목차 1) Git 배경 설명 및 필요성 2) Git 중요 개념 및 사용하기 3) Git 브랜치 관리 기법: Git-flow 4) Git 클라이언트 5) 협업하기: GitHub와 나만의 저장소 만들기

The debate about the correct diffusion model is related to the way to handle the randomness. In this talk, we will see an example which shows that the Stratonovitch interal is the correct way to handle it. The classical kinetic equation is related to the Ito integral. We will construct a new kinetic equation of Stratonovitch type.

In this talk, we will study a triply periodic polyhedral surface whose vertices correspond to the Weierstrass points on the underlying Riemann surface. The symmetries of the surface allow us to construct hyperbolic structures and various translation structures that are compatible with its conformal type. With this explicit data, one can find its algebraic description, automorphism group, Veech group, etc.

We discuss a triangle of viewpoints for circle diffeomorphism groups: analysis, dynamics and group theory. In particular, we illustrate how the regularities (from the analytic side) of diffeomorphisms govern the dynamics and the group theoretical properties of diffeomorphisms. This line of study can be traced back to the works of Hölder, Denjoy, Tsuboi, Thurston and many more.

Haviv (European Journal of Combinatorics, 2019) has recently proved that some topological lower bounds on the chromatic number of graphs are also lower bounds on their orthogonality dimension over R. We show that this holds actually for all known topological lower bounds and all fields. We also improve the topological bound he obtained for the minrank parameter over R – an important graph invariant from coding theory – and show that this bound is actually valid for all fields as well. The notion of independent representation over a matroid is introduced and used in a general theorem having these results as corollaries. Related complexity results are also discussed. This is joint work with Meysam Alishahi.

The Bayesian approach to inverse problems, in which the posterior probabil- ity distribution on an unknown eld is sampled for the purposes of computing posterior expectations of quantities of interest, is starting to become computa- tionally feasible for partial dierential equation (PDE) inverse problems. Bal- ancing the sources of error arising from nite-dimensional approximation of the unknown eld, the PDE forward solution map and the sampling of the prob- ability space under the posterior distribution are essential for the design of ecient computational Bayesian methods for PDE inverse problems. We study Bayesian inversion for a model elliptic PDE with an unknown diusion coef- cient. We consider both the case where the PDE is uniformly elliptic with respect to all the realizations, and the case where uniform ellipticity does not hold, i.e. the coecient can get arbitrarily close to 0 and arbitrarily large as in the log-normal model. We provide complexity analysis of Markov chain Monte Carlo (MCMC) methods for numerical evaluation of expectations under the Bayesian posterior distribution given data, in particular bounds on the overall work required to achieve a prescribed error level. We rst bound the computa- tional complexity of plain MCMC, based on combining MCMC sampling with linear complexity multi-level solvers for elliptic PDEs. The work versus accu- racy bounds show that the complexity of this approach can be quite prohibitive. We then present a novel multi-level Markov chain Monte Carlo strategy which utilizes sampling from a multi-level discretization of the posterior and the for- ward PDE. The strategy achieves an essentially optimal complexity level that is essentially equal to that for performing only one step on the plain MCMC. The essentially optimal accuracy and complexity of the method are mathematically rigorously proven. Numerical results conrm our analysis. This is a joint work with Jia Hao Quek (NTU, Singapore), Christoph Schwab (ETH, Switzerland) and Andrew Stuart (Warwick, England).

Let $F$ be a graph. We say that a hypergraph $\mathcal H$ is an induced Berge $F$ if there exists a bijective mapping $f$ from the edges of $F$ to the hyperedges of $\mathcal H$ such that for all $xy \in E(F)$, $f(xy) \cap V(F) = \{x,y\}$. In this talk, we show asymptotics for the maximum number of edges in $r$-uniform hypergraphs with no induced Berge $F$. In particular, this function is strongly related to the generalized Turán function $ex(n,K_r, F)$, i.e., the maximum number of cliques of size $r$ in $n$-vertex, $F$-free graphs. Joint work with Zoltan Füredi.

Encoder-decoder networks using convolutional neural network (CNN) architecture have been extensively used in deep learning approaches for inverse problems thanks to its excellent performance. However, it is still difficult to obtain coherent geometric view why such an architecture gives the desired performance. Inspired by recent theoretical understanding on generalizability, expressivity and optimization landscape of neural networks, as well as the theory of deep convolutional framelets, here we provide a unified theoretical framework that leads to a better understanding of geometry of encoder-decoder CNNs. Our unified framework shows that encoder-decoder CNN architecture is closely related to nonlinear frame basis representation using combinatorial convolution frames, whose expressivity increases exponentially with the network depth and channels. We also demonstrate the importance of skipped connection in terms of expressivity and optimization landscape. We provide extensive experimental results from various biomedical imaging reconstruction problems to verify the performance encoder-decoder CNNs.

It is a classic result that the maximum weight stable set problem is efficiently solvable for bipartite graphs. The recent bimodular algorithm of Artmann, Weismantel and Zenklusen shows that it is also efficiently solvable for graphs without two disjoint odd cycles. The complexity of the stable set problem for graphs without disjoint odd cycles is a long-standing open problem for all other values of . We prove that under the additional assumption that the input graph is embedded in a surface of bounded genus, there is a polynomial-time algorithm for each fixed . Moreover, we obtain polynomial-size extended formulations for the respective stable set polytopes. To this end, we show that 2-sided odd cycles satisfy the Erdős-Pósa property in graphs embedded in a fixed surface. This result may be of independent interest and extends a theorem of Kawarabayashi and Nakamoto asserting that odd cycles satisfy the Erdős-Pósa property in graphs embedded in a fixed orientable surface. Eventually, our findings allow us to reduce the original problem to the problem of finding a minimum-cost non-negative integer circulation of a certain homology class, which we prove to be efficiently solvable in our case. This is joint work with Michele Conforti, Samuel Fiorini, Gwenaël Joret, and Stefan Weltge.

In this talk, I will discuss some recent developments on the study of singular stochastic wave equations. I also describe some similarities and differences between stochastic wave and heat equations, indicating particular difficulty of the dispersive/hyperbolic problem.

On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. We proved each of these identities under three different interpretations for the double series, and showed that they are intimately connected with the classical circle and divisor problems in number theory. Furthermore, we established many analogues and generalizations of them. This is joint work with Bruce C. Berndt and Alexandru Zaharescu.

In this talk, we present the notion of Stark units in function field arithmetic. This notion was first introduced for investigations on the construction of certain units from the L-function of Drinfeld modules, i.e. log-algebraicity identities. More generally, to a Drinfeld module of higher dimension defined over a function field, we can associate its module of Stark units. We give basic properties of this object and state its connection with Taelman's class formula. Then we will describe the module of Stark units attached to the Carlitz module and its power tensors defined over certain abelian extensions of function fields.

Second day abstract Metamaterials are manmade composite media structured on a scale much smaller than a wavelength. The Minnaert resonance phenomenon makes air bubbles good candidates for the basic building blocks for acoustic metamaterials. Firstly we show the existence of a subwavelength phononic bandgap in bubble phononic crystals, which is proved by an original formula for the quasi-periodic Minnaert resonance frequencies of an arbitrarily shaped bubble. This phenomena can be explained by the periodic inference of cell resonance which is due to the high contrast in both the density and bulk modulus between the bubbles and the surrounding medium. Secondly we show that the bubbly fluid functions like an acoustic metamaterial. Near the Minnaert resonant frequency, an effective medium theory can be derived in the dilute regime. Furthermore, above the Minnaert resonant frequency, the real part of the effective bulk modulus is negative, and consequently the bubbly fluid behaves as a diffusive medium for the acoustic waves. Meanwhile, below the Minnaert resonant frequency, with an appropriate bubble volume fraction, a high contrast effective medium can be obtained, making the subwavelength focusing or superfocusing of waves achievable.

In this talk, we present three related topics on the collective modeling of many-body systems. In the first story, we discuss universal triality relation between bacteria aggregation, Cucker-Smale flocking and Kuramoto synchronization. These three seemingly different phenomena can be integrated into a common nonlinear consensus framework. In our second story, we present a second-order Cucker-Smale modeling on Riemannian manifolds such as the unit circle, the unit sphere in R^3 and Poincare upper half plane model for hyperbolic geometry. Finally, in our third story, we explain how aforementioned collective modeling can be used in the first-order consensus-based optimization algorithm.

First day abstract In acoustics, it is known that air bubbles are subwavelength resonators. Due to the high contrast between the air density inside and outside an air bubble in a fluid, a quasi-static acoustic resonance known as the Minnaert resonance occurs and the bubble behaves as a strong monopole scatterer of sound. Through the application of layer potential techniques and Gohberg–Sigal theory we derive an original formula for the Minnaert resonance frequencies of arbitrarily shaped bubbles. We also provide a mathematical justification for the monopole approximation of scattering of acoustic waves by bubbles at their Minnaert resonant frequency. An acoustic meta-screen is a thin sheet with patterned subwavelength structures, which nevertheless has a macroscopic effect on acoustic wave propagation. When periodic subwavelength bubbles mounted on a reflective surface (with Dirichlet boundary condition) are considered, it is shown that the structure behaves as an equivalent surface with Neumann boundary condition at the Minnaert resonant frequency which corresponds to a wavelength much greater than the size of the bubbles. An analytical formula for this resonance is derived and some numerical simulations confirm its accuracy.

We would give a characterization of those motivic spectra for which the associated slice spectral sequence converges strongly. The characterization is given in terms of the birational covers introduced by the author in order to study the Bloch-Beilinson filtration.

For a graph H , its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions W in L p , p ≥ e ( H ) , denoted by t H ( W ) . One may then define corresponding functionals ∥ W ∥ H := | t H ( W ) | 1 / e ( H ) and ∥ W ∥ r ( H ) := t H ( | W | ) 1 / e ( H ) and say that H is (semi-)norming if ∥ . ∥ H is a (semi-)norm and that H is weakly norming if ∥ . ∥ r ( H ) is a norm. We obtain some results that contribute to the theory of (weakly) norming graphs. Firstly, we show that ‘twisted’ blow-ups of cycles, which include K 5 , 5 ∖ C 10 and C 6 □ K 2 , are not weakly norming. This answers two questions of Hatami, who asked whether the two graphs are weakly norming. Secondly, we prove that ∥ . ∥ r ( H ) is not uniformly convex nor uniformly smooth, provided that H is weakly norming. This answers another question of Hatami, who estimated the modulus of convexity and smoothness of ∥ . ∥ H . We also prove that every graph H without isolated vertices is (weakly) norming if and only if each component is an isomorphic copy of a (weakly) norming graph. This strong factorisation result allows us to assume connectivity of H when studying graph norms. Based on joint work with Frederik Garbe, Jan Hladký, and Bjarne Schülke.

According to a 2018 preprint by Nobuaki Yagita, the conjecture on a relationship between K- and Chow theories for a generically twisted flag variety of a split semisimple algebraic group G, due to the speaker, fails for G the spinor group Spin(17). Yagita's tools include a Brown-Peterson version of algebraic cobordism, ordinary and connective Morava K-theories, as well as Grothendieck motives related to various cohomology theories over fields of characteristic 0. The talk presents a simpler proof using only the K- and Chow theories themselves and, in particular, extending the (slightlymodified) example to arbitrary characteristic.

The Boussinesq abcd system was originally derived by Bona, Chen and Saut [J. Nonlinear. Sci. (2002)] as a rst order 2-wave approximations of the incompressible and irrotational, two dimensional water wave equations in the shallow water wave regime. Among many particular regimes, the Hamiltonian generic regime is characterized by the setting b = d > 0 and a; c < 0. It is known that the system in this regime is globally well-posed for small data in the energy space H1 H1 by Bona, Chen and Saut [Nonlinearity (2004)]. In this talk, we are going to discuss about the decay of small solutions to abcd system in three directions: First, for a weakly dispersive abcd systems (characterized only in terms of parameters a; b and c), all small solutions must decay to zero, locally strongly in the energy space, in proper subset of the light cone jxj jtj. Second, for every ray x = vt, jvj < 1 inside the light cone, small solutions to suciently dispersive system (smallness and dispersion are characterized by v) decay to zero, in proper subset along the ray. Last, small solutions decay to zero in exterior regions jxj jtj under suitable conditions of parameters (a; b; c). All results rule out, among other things, the existence of zero or nonzero speed or super-luminical small solitary waves in each regime where decay is present.

This is joint work with Claudio Munoz.

 

 

We introduce the algebraic connective K-theory and discuss its relations with some other oriented cohomology theories. Then we present recent results on connective K-theory of varieties of parabolic subgroups in semisimple algebraic groups.

For a graph H and an integer k ≥ 1 , the k -color Ramsey number R k ( H ) is the least integer N such that every k -coloring of the edges of the complete graph K N contains a monochromatic copy of H . Let C m denote the cycle on m ≥ 4 vertices. For odd cycles, Bondy and Erd\H{o}s in 1973 conjectured that for all k ≥ 1 and n ≥ 2 , R k ( C 2 n + 1 ) = n ⋅ 2 k + 1 . Recently, this conjecture has been verified to be true for all fixed k and all n sufficiently large by Jenssen and Skokan; and false for all fixed n and all k sufficiently large by Day and Johnson. Even cycles behave rather differently in this context. Little is known about the behavior of R k ( C 2 n ) in general. In this talk we will present our recent results on Ramsey numbers of cycles under Gallai colorings, where a Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles. We prove that the aforementioned conjecture holds for all k and all n under Gallai colorings. We also completely determine the Ramsey number of even cycles under Gallai colorings. Joint work with Dylan Bruce, Christian Bosse, Yaojun Chen and Fangfang Zhang.

In this talk, we study the asymptotic stability of a non-autonomous linear system of ordinary differential equations

Let $G$ be a simply-connected reductive algebraic group over $\mathbb{C}$. For a dominant integral weight $\lambda$, and a reduced decomposition $\mathbf i$ of the longest element in the Weyl group of $G$, the string polytope $\Delta_{\mathbf i}(\lambda)$ is a combinatorial object which encodes weights of $G$-irreducible. It has been observed that the combinatorics of string polytopes depend on a choice of $\mathbf{i}$. Note that string polytopes are non-simple polytopes. Hence they define singular toric varieties. In this talk, we introduce string polytopes when $G = \textrm{SL}_{n+1}(\mathbb{C})$, and we present small resolutions of toric varieties $X_{\Delta_{\mathbf i}(\lambda)}$ for some special $\mathbf i$ using Bott manifolds. This talk is based on joint work with Yunhyung Cho, Yoosik Kim, and Kyeong-Dong Park.

A graph or graph property is ℓ ℓ -reconstructible if it is determined by the multiset of all subgraphs obtained by deleting ℓ ℓ vertices. Apart from the famous Graph Reconstruction Conjecture, Kelly conjectured in 1957 that for each ℓ∈N ℓ∈N , there is an integer n=n(ℓ) n=n(ℓ) such that every graph with at least n n vertices is ℓ ℓ -reconstructible. We show that for each n≥7 n≥7 and every n n -vertex graph G G , the degree list and connectedness of G G are 3 3 -reconstructible, and the threshold n≥7 n≥7 is sharp for both properties.‌ We also show that all 3 3 -regular graphs are 2 2 -reconstructible.

(This is a reading seminar for graduate students.) Recall that there is a spectral sequence strongly converging to the connective $K$-groups whose second page is given by the Zariski cohomology of connective $K$-theory sheaf. In the proof of this result by Quillen, the localization theorem is the most important ingredient. We prove an analogous statement for non-connective $K$-theory with both Zariski and Nisnevich cohomology for noetherian schemes of finite Krull dimension. This theorem is usually phrased as "non-connective algebraic $K$-theory satisfies Zariski and Nisnevich descent". It is known that non-connective algebraic $K$-theory does not satisfy étale descent.

Regularization methods for modeling and prediction are popular in statistics and machine learning. They may be viewed as the procedures that modify the maximum likelihood principle or empirical risk minimization. In particular, methods of regularization defined in reproducing kernel Hilbert spaces (known as kernel methods) provide versatile tools for statistical learning. Primary examples include smoothing splines and support vector machines. I will describe kernel methods focusing on these two examples and discuss some relevant statistical and computational issues. Further I will provide a general description of kernel methods covering mathematical elements and results underlying the methods. Part I: Smoothing Splines (September 30, Monday) Part II: Support Vector Machines (October 2, Wednesday and October 4, Friday) Part III: Kernel Methods (October 8, Tuesday)

A well-known Ramsey-type puzzle for children is to prove that among any 6 people either there are 3 who know each other or there are 3 who do not know each other. More generally, a graph G arrows a graph H if for any coloring of the edges of G with two colors, there is a monochromatic copy of H. In these terms, the above puzzle claims that the complete 6-vertex graph K_6 arrows the complete 3-vertex graph K_3. We consider sufficient conditions on the dense host graphs G to arrow long paths and even cycles. In particular, for large n we describe all multipartite graphs that arrow paths and cycles with 2n edges. This implies a conjecture by Gyárfás, Ruszinkó, Sárkőzy and Szemerédi from 2007 for such n. Also for large n we find which minimum degree in a (3n-1)-vertex graph G guarantees that G arrows the 2n-vertex path. This yields a more recent conjecture of Schelp. This is joint work with Jozsef Balogh, Mikhail Lavrov and Xujun Liu. (*Joint Colloquium between KAIST Mathematical Sciences and IBS Discrete Mathematics Group)

영상 복원(Image restoration, IR)은 low-level vision에서 매우 중요하게 다루는 근본적인 문제 중 하나로서 denoising, deblur, super-resolution 등의 다양한 영상 처리 문제를 포괄한다. 이 발표에서는 영상 복원 분야 중에서도 super-resolution 문제에 대해 집중적으로 다루겠다. 전통적인 model-based optimization 방식과 deep learning을 적용하여 문제를 푸는 방식에 대해, 각각의 장단점과 최신 연구 발전 흐름을 소개한다. 마지막으로는 이 둘을 하나로 잇는 통일된 관점을 제시하고 관련 연구들 살펴본 후, super-resolution 분야에서 아직 남아있는 문제점들을 정리하겠다

We consider an optimal consumption/investment problem to maximize expected utility from consumption. In this market model, the investor is allowed to choose a portfolio that consists of one bond, one liquid risky asset (no transaction costs), and one illiquid risky asset (proportional transaction costs). I fully characterize the optimal consumption and trading strategies in terms of the solution of the free boundary ordinary differential equation (ODE) with an integral constraint. I find an explicit characterization of model parameters for the well-posedness of the problem, and show that the problem is well posed if and only if there exists a shadow price process. Finally, I describe how the investor’s optimal strategy is affected by the additional opportunity of trading the liquid risky asset, compared to the simpler model with one bond and one illiquid risky asset.

Using a simple binomial model, I will present topics in Mathematical Finance (derivative pricing, optimal investment, equilibrium asset pricing). Then we will consider market models with Brownian motion on these topics. Lastly, I will present some result of mine in equilibrium asset pricing in incomplete market setup.

Regularization methods for modeling and prediction are popular in statistics and machine learning. They may be viewed as the procedures that modify the maximum likelihood principle or empirical risk minimization. In particular, methods of regularization defined in reproducing kernel Hilbert spaces (known as kernel methods) provide versatile tools for statistical learning. Primary examples include smoothing splines and support vector machines. I will describe kernel methods focusing on these two examples and discuss some relevant statistical and computational issues. Further I will provide a general description of kernel methods covering mathematical elements and results underlying the methods. Part I: Smoothing Splines (September 30, Monday) Part II: Support Vector Machines (October 2, Wednesday and October 4, Friday) Part III: Kernel Methods (October 8, Tuesday)

This is joint work with Sławomir Kołodziej. We show that the complex m-Hessian
operator of a Holder continuous m-subharmonic function is well dominated by the corresponding capacity. As consequence we obtain the Holder continuous subsolution theorem for the complex m-Hessian equation.

We show finite-time blow up for strong solutions to the 3D Euler equations in two types of corner domains. The first result is for axi-symmetric domains with corners and the data is allowed to be $C^\infty$-smooth if the corner angle is small. In the second case, we utilize the fundamental domain for the octahedral symmetry group. Inside the domain, the data is smooth and can be extended to entire $\mathbb{R}^3$ by a sequence of reflections. In both cases, the solutions have Lipschitz continuous velocity with compact support and have finite energy in particular. This talk is based on joint works with T. Elgindi.

Regularization methods for modeling and prediction are popular in statistics and machine learning. They may be viewed as the procedures that modify the maximum likelihood principle or empirical risk minimization. In particular, methods of regularization defined in reproducing kernel Hilbert spaces (known as kernel methods) provide versatile tools for statistical learning. Primary examples include smoothing splines and support vector machines. I will describe kernel methods focusing on these two examples and discuss some relevant statistical and computational issues. Further I will provide a general description of kernel methods covering mathematical elements and results underlying the methods. Part I: Smoothing Splines (September 30, Monday) Part II: Support Vector Machines (October 2, Wednesday and October 4, Friday) Part III: Kernel Methods (October 8, Tuesday)

We introduce the incompressible Euler equations, which describe the dynamics of volume-preserving inviscid fluids, and describe a few open problems in relation to turbulence. Then we discuss a priori estimates, which give upper bounds on the solution in function spaces. In particular, these estimates guarantee that if the initial fluid velocity is smooth, then there is a unique smooth solution for some time interval. After that, we shall review some recent results towards the opposite direction: attempts in showing lower bounds on the solution instead, with the goal of establishing finite-time singularity formation.

A diffusion equation is one of most famous partial differential equations. Lots of generalized diffusion equations have appeared on the basis of scientific meaning. Equations describing degenerate or unbounded diffusion including stochastic noises are some of them. In this talk, we are going to discuss change of regularity of solutions depending on degeneracy and unboundedness of diffusion and stochastic noise.

Given a graph G , there are several natural hypergraph families one can define. Among the least restrictive is the family B G of so-called Berge copies of the graph G . In this talk, we discuss Turán problems for families B G in r -uniform hypergraphs for various graphs G . In particular, we are interested in general results in two settings: the case when r is large and G is any graph where this Turán number is shown to be eventually subquadratic, as well as the case when G is a tree where several exact results can be obtained. The results in the first part are joint with Grósz and Methuku, and the second part with Győri, Salia and Zamora.